ST3236: Stochastic Process
Tutorial 9
TA: Mar Choong Hock
Email: g0301492@nus.edu.sg
Exercises: 10
Question 1
Messages arrive at a telegraph office as a Poisson
process with mean rate 3 messages per hour
(a)what is the probability that no message arrive
during the morning hours 8:00am to
noon?
(b) what is the distribution of the time at which the
first afternoon message arrives?
Question 1a
P(X(12) - X(8) = 0) = e-4 = e-12
Note that only the duration of the time is required,
property of stationary process.
Is Poisson Process a Markov Process?
Question 1b
Let U1 be the time for the first afternoon message
P(U1  t) = P(X(t) - X(12) = 0) = e-(t-12); t > 12
Thus
FU1(t) = 1 - e-(t-12); t > 12
• The time distribution is exponentially distributed
with mean time = 1/.
• In general, the inter-arrival time between two
events for Poisson process is exponentially
distributed, F(Vk) = 1 - e-(Vk)
(k-1)th arrival
Vk
kth arrival
t
Question 2
Suppose that customers arrive at a facilities
according to a Poisson process having rate  = 2.
Let X(t) be the number of customers that have
arrived up to time t. Determine the following
probabilities and conditional probabilities and
expectations
(a) P(X(1) = 2,X(3) = 6)
(b) P(X(1) = 2|X(3) = 6)
(c) P(X(3) = 6|X(1) = 2)
(d) E{X(1)X(5)[X(3) - X(2)]}
Question 2a
Let  = 2. Make use of the properties of the
counting process and the independent of disjoint
time intervals.
Question 2b
P X (1)  2, X (3)  6 
P X (1)  2 X (3)  6  
P( X (3)  6)
64 6

e
3
e 3 3 6
6!
64 6

e
3
e 6 6 6
6!
2 6 6 . 5 .4 .3 .2


3
66
160

243
Question 2c
P  X (1)  2, X (3)  6 
P X (3)  6 X (1)  2  
P ( X (1)  2)
64 6

e
3
e   2
2!
64 6

e
3
e  2 2 2 2 6 1  4 32  4

 e 
e
2!
3 2
3
Or by counting property,
P(X(3)=6|X(1)=2) = P(X(3)-X(1)=4)
Question 2d
t
1
2
3
5
Question 3
Let X(t) be a Poisson process of rate  = 3 per hour.
Find the conditional probability that there are two
events in the first hour, given that there are five
events in the first three hours.
Question 3
Use Theorem in Lecture notes,
Since we know there are 5 customers in the three
hour interval, we want to know probability of any 2
customers in the 1-hour interval (and 3 customers
in the 2-interval).
Note that where the customers are distributed
uniformly over the time interval, even though the
arrival process is Poisson.
Question 3
From Bayes’ Theorem,
P(X(1)=2|X(3)=5) P(X(3)=5) =
P(X(3)=5|X(1)=2) P(X(1)=2) =
But, by counting property,
P(X(3)=5|X(1)=2)= P(X(3)-X(1) = 3)
Question 4
Customers arrive at a service facility according to a
poisson process of rate  customers/hour.
Let X(t) be the number of customers that have
arrived up to time t. Let W1,W2, … be the
successive arrival times of the customers. Determine
the conditional means E[W5 | X(t) = 4]
and E[W3 | X(t) = 4]
Question 4
Case E[W5 | X(t) = 4]?
In this case, we know that up to time t, the 4th
customer has arrived, what is the average waiting
time s for the 5 customers to arrive?
Question 4
Question 4
Average waiting time for five customer = average
waiting time for four customer + average interarrival time between the fourth and the fifth
customer. (See Q1b)
Question 4 - Optional
W5
V5
W4
1
2
3
4
t
5
E[W5 | X(t) = 4] = E[V5+W4 | W4 = t]
= E[V5+t]
= E[V5] + t
=…
Can try this on the lecture notes problem and
split intervals to W3 + V4 + V5
Question 4
Case E[W3 | X(t) = 4]?
Note: To say the waiting time for 3 customer is less
than s is same as at time s, there is 3 or more
customers arrivals. Because, if at time s there is 3,
then s = W3, if more than 3, s > W3.
Question 4
Case E[W3 | X(t) = 4]?
Note: To say the waiting time for 3 customer is less
than s is same as at time s, there is 3 or more
customers arrivals. Because, if at time s there is 3,
then s = W3, if more than 3, s > W3.
Question 4
Question 5
Let X1(t) and X2(t) be independent Poisson process
having parameters 1 and 2 respectively.
What is the probability that X1(t) = 1 before X2(t) =
1?
That is , what is the probability that first arrival for
process 1 happened earlier than the first arrival for
process 2.
Question 5
Let W1 and W’1 be the waiting time for the first
event in Poisson process 1 and 2 respectively.
Then W1 and W’1 follow exponential distributions
with parameters 1/1 and 1/2 respectively and are
independent of each other. The probability is
Question 5

PW1  W '1   P W1  W '1 W '1  t  p (t )dt

0

 P W1  W '1 W '1  t  2 e 2t dt

0
t


1s 
  1e
ds 2 e 2t dt

0
0
 




1

e

e



 1t
2
 2t
dt
0

 1   2 e 1 2 t dt

0
 1   2 1   2   1 1   2 